| L(s) = 1 | − 2·2-s + 3·4-s − 5-s + 7-s − 4·8-s − 6·9-s + 2·10-s + 4·13-s − 2·14-s + 5·16-s + 3·17-s + 12·18-s − 2·19-s − 3·20-s + 9·23-s + 2·25-s − 8·26-s + 3·28-s + 2·29-s + 2·31-s − 6·32-s − 6·34-s − 35-s − 18·36-s − 14·37-s + 4·38-s + 4·40-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.447·5-s + 0.377·7-s − 1.41·8-s − 2·9-s + 0.632·10-s + 1.10·13-s − 0.534·14-s + 5/4·16-s + 0.727·17-s + 2.82·18-s − 0.458·19-s − 0.670·20-s + 1.87·23-s + 2/5·25-s − 1.56·26-s + 0.566·28-s + 0.371·29-s + 0.359·31-s − 1.06·32-s − 1.02·34-s − 0.169·35-s − 3·36-s − 2.30·37-s + 0.648·38-s + 0.632·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21141604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21141604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1168424766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1168424766\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $D_{4}$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 9 T + 55 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 54 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 14 T + 118 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T + 85 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 21 T + 203 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 111 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 138 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + T + 155 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 62 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 24 T + 318 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.333903366416276869409337985968, −8.295991513667067690108353537790, −7.947884973521772155925676477293, −7.76989229561921659759754167758, −6.98282553857993504844982730977, −6.71387221314145096646563349770, −6.50319096132305285161618030925, −6.18836376813998404455348185981, −5.55101161631916965027399426478, −5.29598192397864335132302771266, −4.89564186070249238724324382005, −4.58932472873564597519375037370, −3.53086768229919273716187311834, −3.40467801706292657044045896623, −3.04154048157613621906731368108, −2.83782174891078903096978056889, −1.84517002538291574365307767787, −1.61804782832537560056523083579, −1.04914165916591187735529366350, −0.13260120595524425536718408737,
0.13260120595524425536718408737, 1.04914165916591187735529366350, 1.61804782832537560056523083579, 1.84517002538291574365307767787, 2.83782174891078903096978056889, 3.04154048157613621906731368108, 3.40467801706292657044045896623, 3.53086768229919273716187311834, 4.58932472873564597519375037370, 4.89564186070249238724324382005, 5.29598192397864335132302771266, 5.55101161631916965027399426478, 6.18836376813998404455348185981, 6.50319096132305285161618030925, 6.71387221314145096646563349770, 6.98282553857993504844982730977, 7.76989229561921659759754167758, 7.947884973521772155925676477293, 8.295991513667067690108353537790, 8.333903366416276869409337985968
